PESHAWAR, Pakistan, June 24 (Reuters) - Gunmen fired on a Pakistan International Airlines plane as it was landing in the northern Pakistani city of Peshawar on Tuesday night, killing a woman on board and injuring three crew members in the third incident at a Pakistani airport this month.

Flight PK 756 was carrying 178 passengers travelling from Saudi Arabia when it came under attack as it was preparing to land, policeman Asghar Khan said at the airport.

The plane was hit by six bullets, police said, killing the Pakistani woman and narrowly missing the captain. At least one bullet struck the plane's engine, police said.

The woman's daughter was sitting next to her when she was shot in the head, PIA official Mohammad Kifayatullah Khan said.

"When I went inside the plane, I saw the woman lying on the seat and her nine-year-old daughter was crying, 'My mother is dead, my mother is dead'," said Khan.

"All the passengers were panicked. Some of them wanted to get out as soon as possible because they were afraid of fire inside the plane.

"The captain of the plane had narrowly escaped," he said. "It would have been a disaster had he been hit."

The incident will raise further questions about whether the government is prepared for a Taliban backlash after officials announced a military operation to flush the militants from their mountain strongholds in North Waziristan on June 15.

Islamabad has promised to tighten security at airports and other potential targets, but critics say decades of neglect of Pakistan's ragged police force has left citizens vulnerable.

On June 8, ten Taliban gunmen attacked the airport in the southern port city of Karachi, Pakistan's financial heart and home to 18 million people. Thirty-four people were killed in the five-hour gunbattle. The Taliban fired on an academy for the security forces at the airport two days later.

On Monday, the government was forced to divert a plane carrying prominent cleric Tahirul Qadri after violence broke out on the ground in Islamabad, with hundreds of supporters armed with sticks battling police, who fired teargas.

The authorities, fearing an escalation of unrest, diverted the plane to the eastern city of Lahore, where Qadri and his supporters refused to leave the plane for hours.

Peshawar's Bacha Khan International Airport has also been a target in the past - in 2012, a Taliban suicide squad staged a car bomb, rocket and gun attack on the airport and nine people, including the five attackers, were killed.

One policeman told Reuters police will not enter the neighbourhood that borders the airport at night for fear of attack by extremists. (Writing by Katharine Houreld; Editing by Sonya Hepinstall)